If you want a solution that does not involve any extra extensions or "high-grade" licences you can try this:

Union A1 and A2 to make A3.

Then select by location where A3 does not have its center in A2 (you may need to select A3 where it DOES have its centre in A2 and then switch selection). OR select by attributes as Union will append attributes from A1 and A2 to A3. This means it is very easy to identify which polygons are not covered by A2 by null attributes that have been acquired from A2.

I am probably late but it may be helpful to others. I was in need of the same functionality in 10.2 that we had in ArcView 3.2, in winch we can cut, union, merge, clip, intersect, and other overlaying operations directly inside the editing session.
I have just found my way by following this procedure (providing you have ArcEditor, as I do and do not know if lower level licenses of ArcMap will let you do this):
What we need:

Polygon feature layer (shp) with polygons to be cut or "bitten"

Cookie cutter polygon

Nbr 2 can be obtained easily: Either you digitize it as a graphic, select it from another feature layer or digitize it in the same or another feature layer.
I have clouds over an area and want to erase this part from a shapefile.

Steps I followed:

Open Edit session in my shapefile (cookie.shp).

select a template and the freehand construction tool (you may have to open these editing windows from Editor toolbar's menu)

Start digitizing around the clouds an overlapping polygon.

convert this new feature to graphics, right-clicking on the layer's name (table of contents) 'Convert features to graphics' (convert selected, draw both graphics and features).

erase the new feature (unless you want it to remain)

convert this graphic to feature (Drawing toolbar) as a new shapefile (cutter.shp)

select the feature in cutter.shp

in Editor Toolbar's menu, select option Clip...

if you are fine with the result, save edits and stop editing.

Beware that this clipping will bite any polygon in cookie.shp that is overlapped by the cutter.

Notice that steps 1 to 7 are to obtain the cutter, so to speak. If you already have your cutter in A2.shp, just select it with the selection tool and perform the biting while editing A1.shp (A1, A2 are your example files).